19. Let $\mu^*$ be an outer measure on $X$ induced from a finite premeasure $\mu_0$. If $E \subset X$, define the inner measure of $E$ to be $\mu_*(E) = \mu_0(X) - \mu^*(E^c)$. Then $E$ is $\mu^*$-measurable iff $\mu^*(E) = \mu_*(E)$. (Use Exercise 18.)

I'm trying to prove that the inner/outer definition of measurability implies the standard definition.

In particular I am having a hard time showing that for an open set $G$ containing $A$, where $A$ satisfies Lebesgue's definition of measurability and the difference in outer measures is $K$, that the outer measure of the difference of the sets is $K$.

  • $\begingroup$ That looks like Folland's Real Analysis, yes? $\endgroup$ – Olorun Jan 5 '16 at 2:08

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